Everyone looks down to first base to find out whether or not the batter actually swung at the pitch.
Checked Swing
The pitch comes in, the batter flinches like they’re going to swing, but they don't fully commit to that swing, and all of a sudden, the umpire and catcher are pointing at things. What’s happened is called a “check swing.” The batter saw a pitch, thought for a second that they wanted to swing, but at the last second, they decided to check themselves. The determination of whether or not they wrecked themselves is left to the umpires.
But first, a question: was the pitch a strike? If the home plate umpire says “yes,” then it doesn't matter that the batter checked their swing. The pitch still goes as a strike. This is also true if the strike call comes via the ABS challenge system. If the pitch was in the strike zone, it's a strike.
Similarly, if the home plate umpire decides that the batter swung, then they will just call a swinging strike, just like any other swinging strike. They may choose to point at the batter as part of the call, as a way of saying “this strike is because you swung.” This, importantly, means that the batter doesn't have any recourse in the matter. The ump says they swung, so they swung.
Ultimately, if the umpire thinks the batter did not swing, and that the pitch was a ball, then they will call a ball. If the catcher thinks that maybe the batter did swing, they can appeal the check-swing part of the call with the first-base umpire (for right-handed batters) or the third-base umpire (for left-handed batters). This is what the pointing is all about; the catcher appeals to the umpire in question, and the home-plate umpire points to indicate that they are granting the appeal. The base umpire then makes their determination, and play continues. The base umpire indicates a swing with a kind of fist-pump motion (similar to the signal for “out”), or by waving their arms with their palms down (similar to the signal for “safe”). Note that batters cannot appeal a check swing call, though you will sometimes see them join in the pointing anyway. The appeal is not like other challenges in that there is no limit to the number of checked-swings one can challenge, and if the catcher loses the appeal, they're not prevented from appealing again, later on. However, the home-plate umpire is under no obligation to grant an appeal, so you won't see catchers pointing every time the batter flinches.
Here’s the Fun Part: there is nothing in baseball’s official rulebook that defines what a swing is, nor anything to define what a check swing is. You might hear announcers talk about things like “breaking the plane,” in which the head of the bat crosses the front part of home plate, but nothing of that is in the actual rules. Which means that the umpire calling the swing is making a pure judgement call, which is always the sort of thing that goes over well, especially in tense situations where a call may mean the difference between extending a plate appearance, or striking out. No, whatever the umpire says, goes1.